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The spin group Spin(n)Spin(n) is the universal covering space of the special orthogonal group SO(n)SO(n). By the usual arguments it inherits a group structure for which the operations are smooth and so is a Lie group like SO(n)SO(n).

For special cases in low dimensions see at: Spin(2), Spin(3), Spin(4), Spin(5), Spin(6), Spin(7), Spin(8)



A quadratic vector space (V,,)(V, \langle -,-\rangle) is a vector space VV over finite dimension over a field kk of characteristic 0, and equipped with a symmetric bilinear form ,:VVk\langle -,-\rangle \colon V \otimes V \to k.

Conventions as in (Varadarajan 04, section 5.3).

We write q:vv,vq\colon v \mapsto \langle v ,v \rangle for the corresponding quadratic form.


The Clifford algebra CL(V,q)CL(V,q) of a quadratic vector space, def. , is the associative algebra over kk which is the quotient

Cl(V,q)T(V)/I(V,q) Cl(V,q) \coloneqq T(V)/I(V,q)

of the tensor algebra of VV by the ideal generated by the elements vvq(v)v \otimes v - q(v).

Since the tensor algebra T(V)T(V) is naturally \mathbb{Z}-graded, the Clifford algebra Cl(V,q)Cl(V,q) is naturally /2\mathbb{Z}/2\mathbb{Z}-graded.

Let ( n,q=||)(\mathbb{R}^n, q = {\vert -\vert}) be the nn-dimensional Cartesian space with its canonical scalar product. Write Cl ( n)Cl^\mathbb{C}(\mathbb{R}^n) for the complexification of its Clifford algebra.


There exists a unique complex representation

Cl ( n)End(Δ n) Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)

of the algebra Cl ( n)Cl^\mathbb{C}(\mathbb{R}^n) of smallest dimension

dim (Δ n)=2 [n/2]. dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]} \,.

The Pin group Pin(V;q)Pin(V;q) of a quadratic vector space, def. , is the subgroup of the group of units in the Clifford algebra Cl(V,q)Cl(V,q)

Pin(V,q)GL 1(Cl(V,q)) Pin(V,q) \hookrightarrow GL_1(Cl(V,q))

on those elements which are multiples v 1v nv_1 \cdots v_{n} of elements v iVv_i \in V with q(v i)=1q(v_i) = 1.

The Spin group Spin(V,q)Spin(V,q) is the further subgroup of Pin(V;q)Pin(V;q) on those elements which are even number multiples v 1v 2kv_1 \cdots v_{2k} of elements v iVv_i \in V with q(v i)=1q(v_i) = 1.

Specifically, “the” Spin group is

Spin(n)Spin( n). Spin(n) \coloneqq Spin(\mathbb{R}^n) \,.

A spin representation is a linear representation of the spin group, def. .



By definition the spin group sits in a short exact sequence of groups

2SpinSO. \mathbb{Z}_2 \to Spin \to SO \,.

Relation to Whitehead tower of orthogonal group

The spin group is one element in the Whitehead tower of O(n)O(n), which starts out like

Fivebrane(n)String(n)Spin(n)SO(n)O(n). \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

The homotopy groups of O(n)O(n) are for kk \in \mathbb{N} and for sufficiently large nn

π 8k+0(O) = 2 π 8k+1(O) = 2 π 8k+2(O) =0 π 8k+3(O) = π 8k+4(O) =0 π 8k+5(O) =0 π 8k+6(O) =0 π 8k+7(O) =. \array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

By co-killing these groups step by step one gets

cokillthis toget π 0(O) = 2 SO π 1(O) = 2 Spin π 2(O) =0 π 3(O) = String π 4(O) =0 π 5(O) =0 π 6(O) =0 π 7(O) = Fivebrane. \array{ cokill\, this &&&& to\,get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.

Via the J-homomorphism this is related to the stable homotopy groups of spheres:

Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

Exceptional isomorphisms

In low dimensions the spin groups happens to be isomorphic to various other classical Lie groups. One speaks of exceptional isomorphisms or sporadic isomorphisms.

See for instance (Garrett 13). See also division algebra and supersymmetry.

In the following Sp(n)Sp(n) denotes the quaternionic unitary group in quaternionic dimension nn.

We have

  • in Euclidean signature

    • Spin(1)O(1)Spin(1) \simeq O(1)

    • Spin(2)U(1)SO(2)S 1\simeq U(1) \simeq SO(2) \simeq S^1 (SO(2), the circle group, see also at real Hopf fibration)

      the projection Spin(2)SO(2)Spin(2)\to SO(2) corresponds to S 12S 1S^1\stackrel{\cdot 2}{\longrightarrow} S^1, see also at Theta characteristic

    • Spin(3)Sp(1)SU(2)S 3\simeq Sp(1) \simeq SU(2) \simeq S^3 (the special unitary group SU(2)

      the inclusion Spin(2)Spin(3)Spin(2) \hookrightarrow Spin(3) corresponds to the canonical S 1S 3S^1 \hookrightarrow S^3 (see e.g. Gorbunov-Ray 92)

    • Spin(4)Sp(1)×Sp(1)S 3×S 3\simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3

      this is given by identifying 4\mathbb{R}^4 \simeq \mathbb{H} with the quaternions and SU(2)S 3SU(2) \simeq S^3 with the group of unit quternions. Then left and right quaternion multiplication gives a homomorphism

      SU(2)×SU(2)SO(4) SU(2) \times SU(2) \longrightarrow SO(4)
      (g,h)(xg 1xh) (g,h) \mapsto ( x \mapsto \; g^{-1} x h )

      which is a double cover and hence exhibits the isomorphism.

      In particular therefore the inclusion Spin(3)Spin(4)Spin(3) \hookrightarrow Spin(4) corresponds to the diagonal S 3S 3×S 3S^3 \hookrightarrow S^3 \times S^3.

      At the level of Lie algebras 𝔰𝔬(4) 2 4\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4 and the ±1\pm 1-eigenspaces of the Hodge star operator : 2 4 4\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4 gives the direct sum decomposition 𝔰𝔬(4)𝔰𝔲(2)𝔰𝔲(2)𝔰𝔬(3)𝔰𝔬(3)\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)

    • Spin(5)Sp(2)\simeq Sp(2) (an indirect consequence of triality, see e.g. Čadek-Vanžura 97)

    • Spin(6)SU(4)\simeq SU(4) (the special unitary group SU(4))

  • in Lorentzian signature

  • in anti de Sitter signature

    • Spin(2,2)SL(2,)×SL(2,)Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})

    • Spin(3,2)Sp(4,)Spin(3,2) \simeq Sp(4,\mathbb{R})

    • Spin(4,2)SU(2,2)Spin(4,2) \simeq SU(2,2)

  • in mixed signature

Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

exceptional spinors and real normed division algebras

AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string


rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group

see also


Spin geometry

See spin geometry

In physics

The name arises due to the requirement that the structure group of the tangent bundle of spacetime lifts to Spin(n)Spin(n) so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)

See spin structure.

The Whitehead tower of the orthogonal group looks like

\cdots \to fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,
superconformal group


Textbook accounts:

See also

Examples of sporadic (exceptional) spin group isomorphisms incarnated as isogenies onto orthogonal groups are discussed in

The exceptional isomorphism Spin(5) \simeq Sp(2) is discussed via triality in

Discussion of the cohomology of the classifying space BSpinB Spin includes

  • E. Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.

  • Harsh Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)

Last revised on October 31, 2023 at 17:00:19. See the history of this page for a list of all contributions to it.